The <orbit-cone correspondence> is an inclusion-reversing bijection
$$
\{\sigma\in\Sigma\}\longleftrightarrow
\{T_N\text{-orbits in }X_\Sigma\},
\qquad \sigma\longmapsto O(\sigma).
$$
If $n=\operatorname{rank}N$, then
$$
\dim O(\sigma)=n-\dim\sigma,
\qquad
O(\sigma)\cong\operatorname{Hom}(M\cap\sigma^\perp,\mathbb C^\times).
$$
The affine chart and <orbit closure in a toric variety> are
$$
U_\sigma=\bigcup_{\tau\preceq\sigma}O(\tau),
\qquad
V(\sigma)=\overline{O(\sigma)}
=\bigcup_{\tau\supseteq\sigma}O(\tau).
$$
In particular, the zero cone corresponds to the dense <algebraic torus>, while maximal cones correspond to torus-fixed points.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 152 / 1 / b / Solution

{title2=$V(\sigma)$}

For $\sigma\in\Sigma$, the closure of its torus orbit is
$$
V(\sigma)=\overline{O(\sigma)}
=\bigcup_{\tau\supseteq\sigma}O(\tau).
$$
Thus orbit-closure inclusion reverses face inclusion.


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